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I must be wrong here, but still want to know where I am wrong. I found the data of Libor rate and swap rate from this link: http://www.interestrateswapstoday.com/libor-rates.html

At the time, I read

USD 6-month Libor rate: L6 = 2.8858%

USD 12-month Libor rate: L12 = 3.1006%

USD 1-year Swap rate: S1 = 2.828%

Then, I calculate

6-month zero bond: P6 = 1/(1 + L6*0.5) = 0.9857762347093784

12-month zero bond: P12 = 1/(1+ L12) = 0.9699264601757894

If above 1-year swap rate is semi-annually settled (this is what I understood, but did not see official explanations), then one shall have

1 = S1*0.5*P6 + (S1*0.5+1)*P12

But, the right hand side is equal to 0.9975800962814657, strictly less than 1. Does it mean there is slight arbitrage opportunity, or otherwise I misunderstood the definition of the rates in the above?

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    $\begingroup$ The calculations you performed were standard before the financial crisis. Since the crisis, people have come to realize that LIBOR is not a risk-free rate, invalidating these calculations. Simply put, 6m, 12m, and 1y LIBORs embed different credit risks and do not belong to the same curve. Try searching for multi-curve; there are many great discussions over the past decade. $\endgroup$ – Helin Dec 12 '18 at 2:18
  • $\begingroup$ Thanks for your comment, I will search for the multi-curve $\endgroup$ – kenneth Dec 12 '18 at 7:28
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You have discovered what is called the tenor basis. This is where theoretical finance and empirical finance part ways, at least a little bit.

12 month LIBOR means the rate at which I will give someone (AA credit - most likely another bank) a cash loan. I will send out the money and get it back 12 months later. If the borrower defaults then I'm in line with other unsecured creditors.

The 1 year swap rate is the simply the fixed rate for a swap vs 3 month libor. Very little cash goes back and forth unless the market moves.

They are two totally different transactions.

The 12 month rate is almost always higher than the implied rate from the 3 month ED futures. There are various theories about why that should be. Some people believe that it reflects the risk of a longer loan and longer credit exposure.

Fundamentally there is , sort of, an arbitrage. If you are money center bank you could lend to someone for 12 months and finance that by borrowing from ... depositors, other banks, etc ... for shorter terms. Roll it and make money. Hedge the rate exposure with 1y swaps.

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More specifically the front end of the swap curve is constructed from 3m eurodollar futures and not the 6m, 12m libor. The credit crisis did help make that change. It's not because lIBOR was no longer believed to be a risk free rate as it's always been a bank funding rate essentially. It is because a vanilla libor 1yr swap is a derivative of 3m libor since that is the floating leg used to solve for the fixed leg. 12m libor and a 1yr swap are similar but they're different in that a 12m libor loan is a term loan and a 1yr swap is a revolving loan and in the crisis a 3m loan, or a series of 4 3 month loans, became a lot different than a 1yr loan.

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Perhaps, but your result could be so close to 1.0000 that exact time each observation is gathered could be a factor. There could be a difference in a matter of seconds or minutes among when these numbers are collected.

Also, your S1 is 3 digits after the decimal and the others have 4. This minor detail could also be cause of it being almost 1.0000 but not exactly.

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